Mathematics

The coincidental down-degree expectations (CDE) phenomenon

Sam Hopkins   [email] (University of Minnesota)

Abstract: Let P be a finite poset. Consider two probability distributions on P: the uniform distribution; and the distribution where each element occurs with probability proportional to the number of maximal chains passing through it. Reiner, Tenner, and Yong observed that for many posets P of interest to algebraic combinatorialists, the expected Hasse diagram down-degree is the same for both these distributions. They called this the "coincidental down-degree expectations" (CDE) property for posets. I will review the history of the study of CDE posets: its origins in the algebraic geometry of curves, its connections to the tableaux/symmetric function/Schubert calculus world, etc. I will also present the "toggle perspective," which has been useful in establishing that various distributive lattices are CDE. Time permitting I will explain a forthcoming proof of a conjecture of Reiner-Tenner-Yong which says that certain weak order intervals corresponding to vexillary permutations are CDE. The proof involves expanding the "toggle perspective" to the setting of semidistributive lattices (following Barnard and Thomas-Williams).

Surreal Substructures

Vincent Bagayoko (Paris 7)

Abstract: We study the class No of surreal numbers equipped with its natural order and simplicity relation. A "surreal substructure" of No is a subclass that is isomorphic to No itself for the restricted order and simplicity relation. Such surreal substructures frequently arise when studying various operations on the surreals such as algebraic operations, exponentiation, or infinite summation. This makes it worthwhile to investigate the general properties of surreal substructures in their own right. We will give their general properties, provide many examples, and focus on two applications of surreal substructures: to produce fixed points of certain specific surreal functions and to define classes of surreal numbers defined as simplest in a convex class of No. We will mention their use in studying the hyperserial structure of No conjectured by van der Hoeven.

The sum-product problem

George Shakan (Illinois Math)

Abstract: The Erdős–Szemerédi sum-product problem asserts that for any $A$ in the integers either $|A+A|$ or $|AA|$ is at least $|A|^2$ up to an arbitrarily small power of $|A|$. In this talk, we'll discuss recent progress and further questions.

Orientation data for coherent sheaves on local $\mathbb{P}^2$

Yun Shi (UIUC Math)

Abstract: Orientation data is an ingredient in the definition of Motivic Donaldson-Thomas (DT) invariant. Roughly speaking, it is a square root of the virtual canonical bundle on a moduli space. It has been shown that there is a canonical orientation data for the stack of quiver representations for a quiver with potential. In this talk, I will briefly introduce Motivic DT invariant, and the role of orientation data in its definition. I will then give a construction of orientation data for the stack of coherent sheaves on local $\mathbb{P}^2$ based on the canonical orientation data from quiver representations.